This project will examine new methodology for Bayesian inference and model selection with censored failure time data and longitudinal data. In particular, we examine univariate and bivariate survival models with a surviving (cure) fraction and propose several novel methods for inference and computations. In addition, we develop Bayesian methodology for longitudinal data models, including random effects model and time series models. The methodology addresses problems occurring frequently in clinical investigations for chronic disease, including cancer and AIDS, as well as problems relating to the environment. The specific objectives of the project are to: 1) Develop and study Bayesian methods of inference and model selection for survival models with a surviving fraction. In particular, we will: i) develop a new Bayesian univariate model with a surviving fraction. We will examine the theoretical properties of the model and examine the inclusion of covariates. We will propose a class of informative prior distributions for the regression coefficients based on historical data, examine their theoretical properties, develop model selection tools, and propose new computational Markov chain Monte Carlo (MCMC) algorithms for inference. We will also develop EM algorithms for obtaining maximum likelihood estimates for the parameters. ii) develop extensions of (i) to bivariate survival models with a cure fraction. Specifically, we propose a new bivariate survival cure rate model and examine its theoretical properties. Inclusion of covariates is developed for this model. Informative prior distributions based on historical data will be proposed and their properties will be studied. Model selection strategies will be proposed. Novel MCMC computational methods will be proposed and implemented. 2) Develop and study Bayesian methods of inference for models for longitudinal data. Specifically, we will examine Bayesian methods fo estimation, model selection, and computation for i) random effects models and ii) time series models. For each of these types of models, we will propose a class of informative prior distributions and study their theoretical properties. We will also develop novel selection methodology, including the development of a priors based on historical data for the parameters as well as the model space. We will also develop efficient MCMC computational models for computing model probabilities and Bayes factors.